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JEE Main 20252 AprilEvening Shift
Exam
JEE Main
Year
2025
Date
2 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Statistics

JEE Main 2025 · 2 April · Evening Shift · Mathematics · Statistics

If the mean and the variance of 6,4, a, 8, b, 12,10 , 13 are 9 and...

If the mean and the variance of 6,4,a,8,b,12,106,4, a, 8, b, 12,10, 13 are 9 and 9.25 respectively, then a+b+aba+b+a b is equal to :

Options

  1. A105105
  2. B103103
  3. C100100
  4. D106106

Correct answer

B. 103103

Step-by-step solution

∵ mean =9∴53+a+b=72⇒a+b=19∵σ2=374 and (X‾)2+σ2=∑x12N⇒81+374=529+a2+b28⇒648+74=529+a2+b2⇒a2+b2=193∵a+b=19⇒a2+b2+2ab=361⇒2ab=168⇒ab=84∴a+b+ab=103\begin{aligned} & \because \text{ mean }=9 \\ & \therefore 53+\mathrm{a}+\mathrm{b}=72 \\ & \Rightarrow \mathrm{a}+\mathrm{b}=19 \\ & \because \sigma^2=\frac{37}{4} \text{ and }(\overline{\mathrm{X}})^2+\sigma^2=\frac{\sum \mathrm{x}_1^2}{\mathrm{ N}} \\ & \Rightarrow 81+\frac{37}{4}=\frac{529+\mathrm{a}^2+\mathrm{b}^2}{8} \\ & \Rightarrow 648+74=529+\mathrm{a}^2+\mathrm{b}^2 \\ & \Rightarrow \mathrm{a}^2+\mathrm{b}^2=193 \\ & \because \mathrm{a}+\mathrm{b}=19 \Rightarrow \mathrm{a}^2+\mathrm{b}^2+2 \mathrm{ab}=361 \\ & \Rightarrow 2 \mathrm{ab}=168 \\ & \Rightarrow \mathrm{ab}=84 \\ & \therefore \mathrm{a}+\mathrm{b}+\mathrm{ab}=103\end{aligned}
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